Fill in the blank. In the diagram, we can conclude
that 𝐿 sub one is what 𝐿 sub two.
In the diagram, we can observe that
there are four different lines which intersect at various points. The two lines in particular that we
need to consider are the lines 𝐿 sub one and 𝐿 sub two, which are marked here in
pink. The missing word or words in the
statement will be something that describes the relationship between these two
lines. It might be useful to consider what
possible relationships two lines in two dimensions could have.
So let’s begin with the word
“perpendicular.” Perpendicular lines cross at right
angles. Or the lines could be parallel. That would mean they would always
be the same distance apart and never meet. Next, we could also describe two
lines as simply intersecting. So the lines will meet or cross,
but not necessarily at right angles. Finally, another possible option is
that the lines are coincident. That means they would share all
points and would in fact lie on top of one another.
By looking at the diagram, we could
rule out two of these possible options, because 𝐿 sub one and 𝐿 sub two don’t
cross at 90 degrees and they appear as two distinct lines. Notice that if two lines are not
parallel, then they would be intersecting at some point, even if we don’t see it on
the diagram. The most likely situation is that
these two lines are parallel. But we can’t just give this as the
answer without proving it in some way.
Let’s recall the theorem that if
corresponding angles, alternate interior angles, or alternate exterior angles formed
by a transversal cutting two lines are congruent, then the lines cut by the
transversal are parallel. So let’s see what we can tell from
the given angle measures in the figure.
We are given three angle
measures. But none of these are equal. So we’ll have to see if there are
any other angle measures we can determine. One possible option would be to see
if we can find the measure of this angle marked in blue, since it would potentially
be an alternate interior angle to the angle whose measure is given as 100
degrees. We can define this unknown angle
measure as 𝑥 degrees.
Now, consider that the angle
measure of 𝑥 degrees lies within this triangle with another angle of measure 30
degrees and a third unknown angle, which we can call 𝑦 degrees. The angle measures of 𝑦 degrees
and 130 degrees lie on a straight line. So these angle measures must sum to
180 degrees, which means that 𝑦 degrees is equal to 50 degrees.
We can now use the property that
the sum of the interior angle measures in a triangle is 180 degrees to help us find
the value of 𝑥. The three angle measures of 30
degrees, 50 degrees, and 𝑥 degrees must sum to 180 degrees. And by simplifying the left-hand
side and subtracting 80 degrees from both sides, we have that 𝑥 degrees is 100
Now, remember, we wanted to find
the value of this angle measure because we wanted to check if there are congruent
angles. And we have found that a pair of
alternate angles formed by a transversal cutting two lines are congruent. Therefore, the two lines are
parallel. So we can complete the blank with
the words “parallel to,” since we have proved that in the diagram, line 𝐿 sub one
is parallel to line 𝐿 sub two.